ENERGY 1; Denoting the kinetic energy by T and putting the defii] tion into analytical language we obtain r-- rt-m'- l r^dv Jo dt Therefore the kinetic energy of a particle equals one-half the prc net of the mass by the square of its velocity. Since both m ai v2 are positive, kinetic energy must be a positive magnituc The kinetic energy of a system of particles, therefore, eqm the arithmetic sums of the kinetic energies of the individu particles. Thus T = When all the particles of the system have the same veloci where M is the total mass of the system. 154. Work Done in Increasing the Velocity of a Particle. — the velocity of a particle is increased from VQ to v then t work done against the kinetic reaction equals the incre£ in the kinetic energy of the particle. This will be seen frc the following analysis : ™ — ml J Vn at vdv ' »0 = 2" WVD — 2 "WQ I /T- = T-T0. j ^ * The first negative sign indicates the fact that T is the work done agai and not by the kinetic reactions. The second negative sign belongs to the kinetic reaction as it was explai in Chapter VI.